Invertible generative models for inverse problems: mitigating representation error and dataset bias

TL;DR

This paper demonstrates that invertible neural networks, which have zero representation error, serve as effective priors for inverse imaging problems, outperforming traditional generative models and sparsity priors in accuracy and robustness.

Contribution

It introduces the use of invertible neural networks as priors for inverse problems, highlighting their advantages over GANs and sparsity models, with theoretical and empirical validation.

Findings

Invertible priors outperform GAN priors in accuracy for natural images.

Invertible models yield better reconstructions for out-of-distribution images.

Theoretical bounds on recovery error are established for linear invertible models.

Abstract

Trained generative models have shown remarkable performance as priors for inverse problems in imaging -- for example, Generative Adversarial Network priors permit recovery of test images from 5-10x fewer measurements than sparsity priors. Unfortunately, these models may be unable to represent any particular image because of architectural choices, mode collapse, and bias in the training dataset. In this paper, we demonstrate that invertible neural networks, which have zero representation error by design, can be effective natural signal priors at inverse problems such as denoising, compressive sensing, and inpainting. Given a trained generative model, we study the empirical risk formulation of the desired inverse problem under a regularization that promotes high likelihood images, either directly by penalization or algorithmically by initialization. For compressive sensing, invertible priors can yield higher accuracy than sparsity priors across almost all undersampling ratios, and due to their lack of representation error, invertible priors can yield better reconstructions than GAN priors for images that have rare features of variation within the biased training set, including out-of-distribution natural images. We additionally compare performance for compressive sensing to unlearned methods, such as the deep decoder, and we establish theoretical bounds on expected recovery error in the case of a linear invertible model.